Abstract
Compressed sensing is a technique for recovering an unknown sparse signal from a small number of linear measurements. When the measurement matrix is random, the number of measurements required for perfect recovery exhibits a phase transition: there is a threshold on the number of measurements after which the probability of exact recovery quickly goes from very small to very large. In this work we are able to reduce this threshold by incorporating statistical information about the data we wish to recover. Our algorithm works by minimizing a suitably weighted ℓ1-norm, where the weights are chosen so that the expected statistical dimension of the corresponding descent cone is minimized. We also provide new discrete-geometry-based Monte Carlo algorithms for computing intrinsic volumes of such descent cones, allowing us to bound the failure probability of our methods.
Original language | English |
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Pages (from-to) | 486-504 |
Number of pages | 19 |
Journal | Applied and Computational Harmonic Analysis |
Volume | 45 |
Issue number | 3 |
DOIs | |
State | Published - Nov 2018 |
Externally published | Yes |
Keywords
- Compressed sensing
- Intrinsic volumes
- Monte Carlo algorithm
- Statistical dimension
- Weighted ℓ-norm